BUNTES CM abelian varieties

Section 8.1 CM abelian varieties

Let $k$ be a field.

Recall that an abelian variety is a proper group variety over $k\text{.}$ Let $A/k$ be an abelian variety.

Definition 8.1.1. Endomorphism algebra.

The endomorphism ring of $A$ is the ring of all isogenies $A \to A$

\begin{equation*} \End(A) = \Hom_{k-isog}(A,A) \end{equation*}

the endomorphism algebra is

\begin{equation*} \End^0(A) = \End(A) \otimes \QQ \end{equation*}

this is a possibly non-commutative semisimple $\QQ$-alg.

for a semisimple algebra the reduced degree is defined by decomposing

\begin{equation*} B = \prod B_i \end{equation*}

simple algebras with center $k_i\text{.}$

\begin{equation*} [B: k]_{\mathrm{red}}=\sum_{i}\left[B_{i}: k_{i}\right]^{1 / 2} \cdot\left[k_{i}: k\right] \end{equation*}

We can bound the dimension of this algebra by observing that it acts faithfully on the homology / tate module for $\ell \ne \characteristic k\text{,}$ these are dimension $2\dim A$). With Artin-Wedderburn this gives

\begin{equation*} 2\dim A \ge [ \End^0(A) : \QQ] _{red} \ge [ E:\QQ] \end{equation*}

for any etale algebra $E$ in $\End^0(A)\text{.}$

If the first inequality is an equality they both are and we say that $A$ has CM. In this case $\End^0(A)$ is a product of matrix algebras over fields.

Example 8.1.2. Elliptic curves.

We have several possibilities

\begin{equation*} A\colon y^2+ y = x^3- x^2/\QQ \end{equation*}
has $\End^0(A)= \QQ\text{,}$ dim $\sqrt 1 \cdot 1 \le 2$ no CM
\begin{equation*} A\colon y^2 = x^3+ 1/\QQ(\zeta _3) \end{equation*}
has $\End^0(A)= \QQ(\zeta _3)\text{,}$ dim $\sqrt 1 \cdot 2 \le 2\text{,}$ CM, own maximal etale.
\begin{equation*} A\colon y^2 + y = x^3 + x^2 +x + 1/\FF_4 \end{equation*}
we can find 24 automorphisms, that make the group $\SL_2(\FF_3)\text{.}$ And $\End^0(A)$ is the quaternion algebra $\QQ$ ramified at $2,\infty \text{.}$ So
\begin{equation*} [\End^0(A) : \QQ]_{red} = \sqrt 4 \cdot 1 = 2 \end{equation*}
here the maximal etale algebras inside are the imaginary quadratic fields contained in this quaternion algebra.
\begin{equation*} \left(\frac{-1,-1}{\QQ}\right) \end{equation*}
The same example over $\FF_2\text{,}$ of the 24 automorphisms only 2 are defined over $\FF_2\text{,}$ and we have
\begin{equation*} \End^0(A) = \QQ(\sqrt{-2}) \end{equation*}

\begin{equation*} [\End^0(A) : \QQ]_{red} = \sqrt 1 \cdot 2 = 2 \end{equation*}
so CM again with one of the same etale algebras $E$ as before.
Given a CM elliptic curve $A/\QQ$ with CM by $F$ can take the product
\begin{equation*} A \times A / k \end{equation*}
this has dimension two and
\begin{equation*} \End^0(A) = \Mat_{2\times 2}(F) \end{equation*}
this is a 4-dim algebra over its center of dim 2
\begin{equation*} \sqrt 4 \cdot 2 = 4 = 2 \dim A \end{equation*}
etale algebra
\begin{equation*} E = F \times F\text{.} \end{equation*}
Given non-isogenous CM elliptic curves $A,A'/\QQ$ with CM by $F,F'$ can take the product
\begin{equation*} A \times A' / k \end{equation*}
this has dimension two and
\begin{equation*} \End^0(A) = F \times F' \end{equation*}
this is a product of two 1-dimensional algebras over their centers
\begin{equation*} \sqrt 1 \cdot 2 +\sqrt 1 \cdot 2 = 4 = 2 \dim A \end{equation*}
etale algebra
\begin{equation*} E = F \times F'\text{.} \end{equation*}

Subsection 8.1.1 Construction over $\CC$

We can construct many examples over $\CC$ as follows.

Definition 8.1.3. CM-pairs.

A CM-pair is a pair

\begin{equation*} E, \Phi \end{equation*}

where $E$ is a product of CM fields (aka a CM-algebra). Such an algebra has an involution

\begin{equation*} \iota _E \colon E\to E \end{equation*}

non-trivial on each field such that for any embedding

\begin{equation*} \phi \in \Hom(E, \CC) \end{equation*}

\begin{equation*} \phi \circ \iota _E = \bar \cdot \circ \phi \text{.} \end{equation*}

$\Phi $ is a CM-type

\begin{equation*} \Phi \subset \Hom(E, \CC) \end{equation*}

of cardinality $\dim E/2$ s.t

\begin{equation*} \iota _E \Phi \cup \Phi = \Hom(E, \CC)\text{.} \end{equation*}

Given such a CM-pair and a choice of lattice

\begin{equation*} \Lambda \subseteq E \end{equation*}

we can form a complex torus

\begin{equation*} \CC^\Phi / \Phi (\Lambda )\text{.} \end{equation*}

To make this into an abelian variety we need the existence of a polarization. The relevant Riemann forms are given by

\begin{equation*} E\times E \to \QQ \end{equation*}

\begin{equation*} (x,y ) \mapsto \trace_{E/\QQ}(\alpha x \iota _E( y)) \end{equation*}

for $\alpha \in E^\times$ satisfying

\begin{equation*} \iota _E \alpha = -\alpha \end{equation*}

\begin{equation*} \im(\phi (\alpha )) \gt 0,\,\forall \phi \in \Phi \text{.} \end{equation*}

So we can make a choice of $\alpha $ and obtain an abelian variety in this way. Such abelian varieties have CM as

\begin{equation*} \End^0(A) \end{equation*}

contains etale algebra $E$ which has dimension $2 \cdot \# \Phi = 2 \dim A\text{.}$

Theorem 8.1.4. Tate.

Every abelian variety over a finite field has CM.

Theorem 8.1.5. Grothendieck.

Every abelian variety with CM over an algebraically closed field $K$ of characteristic $p$ is isogenous to a CM abelian variety over a finite field.

Over $\CC\text{:}$ A simple abelian variety has CM iff $\End^0(A)$ is a field of dimension $2 \dim A\text{,}$ moreover such a field is necessarily a CM field.

Proposition 8.1.6.

Let $k \subseteq \CC$ be algebraically closed them

\begin{equation*} \{\text{abvar }/k\} \to \{\text{abvar }/k\} \end{equation*}

is fully faithful and the essential image contains all CM abelian varieties.

Proof.

(Sketch) Full faithfulness follows from: The torsion points are algebraic and Zariski dense. For essential image take $A$ we can find $A' /k$ with same CM-type by spreading out type stuff, so $A'_\CC$ is isogenous to the original. Now the kernel of the isogeny is algebraic again so can quotient by it in both categories.

So CM abvars $/k$ are equivalent to CM abvars $/\CC\text{.}$

Using Neron(-Ogg-Shafarevich) we get

Proposition 8.1.7.

Let $A$ be an abelian variety over a number field $k$ with complex multiplication. Then $A$ has potential good reduction at all finite primes of $k\text{.}$

Let $A$ be an abelian variety with complex multiplication by $E$ over a field $k,$ and let $\mathfrak{a}$ be a lattice ideal in $R .$ A surjective homomorphism $\lambda^{\mathfrak{a}}: A \rightarrow A^{\mathfrak{a}}$ is an a-multiplication if every homomorphism $a: A \rightarrow A$ with $a \in \mathfrak{a}$ factors through $\lambda^{\mathfrak{a}},$ and $\lambda^{\mathfrak{a}}$ is universal for this property, in the sense that, for every surjective homomorphism $\lambda^{\prime}: A \rightarrow A^{\prime}$

BUNTES: BU Number Theory Expository Seminar